For example, if you are handed several pieces of fruit, you could divide the fruit according to species, or according to color, or according to size, or according to approximate number of seeds. "Classification" is the operation of examining an object and determining what variety it is according to the criteria of interest, and each classification operation divides the objects in to classes (distinct kinds of entities under the classification criteria)
Note from the above that an object does not inherently have a distinct class, only a class according to some criteria of interest (e.g., species, color).
It is common in classification problems to encounter objects that are not clearly one particular class (e.g., where exactly does "red" end and "orange" begin if you are using color). It is also common in classification problems to encounter objects that do not appear to be any of the pre-defined classes.
There is also an important branch of classification in which one is given a number of measurements for each object, and asked to find some combination of measurements and cut-off values to sort the objects in to distinct kinds (classes) without knowing ahead of time which of the measurements or measurement values are important -- possibly without even knowing how many different kinds of objects are present. For example, the measurements might be biochemical concentrations from blood tests, and one might be attempting to determine which of the patients had (say) yaws, without knowing ahead of time exactly how that disease tends to change blood chemistry.
